L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 4·11-s − 6·13-s − 2·14-s + 16-s + 4·17-s + 19-s + 4·22-s − 6·26-s − 2·28-s + 10·29-s − 2·31-s + 32-s + 4·34-s + 2·37-s + 38-s − 8·41-s + 8·43-s + 4·44-s − 3·49-s − 6·52-s − 6·53-s − 2·56-s + 10·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.20·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.852·22-s − 1.17·26-s − 0.377·28-s + 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.162·38-s − 1.24·41-s + 1.21·43-s + 0.603·44-s − 3/7·49-s − 0.832·52-s − 0.824·53-s − 0.267·56-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.085388708\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.085388708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55385536858252009827497182117, −6.95202853421741408871651283602, −6.42543149678495762340368952550, −5.71251830782427962545947506738, −4.89961596325054362357107947799, −4.34145866528188736551545243324, −3.39228050924654068416251697139, −2.90727030677509597771969141650, −1.91472879079515969689388728020, −0.76864370346727157384301943155,
0.76864370346727157384301943155, 1.91472879079515969689388728020, 2.90727030677509597771969141650, 3.39228050924654068416251697139, 4.34145866528188736551545243324, 4.89961596325054362357107947799, 5.71251830782427962545947506738, 6.42543149678495762340368952550, 6.95202853421741408871651283602, 7.55385536858252009827497182117